DifferentialGeometry/Tensor/BivectorSolderForm - Maple Help

Tensor[BivectorSolderForm] - construct the bivector solder form defined by a solder form

Calling Sequences

BivectorSolderForm(sigma, spinorType, indexlist)

Parameters

sigma      - a solder form

spinorType - a string, either "spinor" or "barspinor"

indexlist  - (optional) the keyword argument indexlist = ind, where ind is a list of 4 index types "con" or "cov"

Description

 • A bivector is a skew-symmetric, rank 2 contravariant tensor. On a 4-dimensional manifold with solder form $\mathrm{σ}$ there is a 1-1 correspondence between bivectors and symmetric rank 2 spinors.  This correspondence is explicitly furnished by the bivector solder forms and $\stackrel{‾}{S}$ which are defined in terms of the solder form s by

and

.

 • The tensor indices of the bivector solder forms are raised and lowered with the metric $g$ defined by $\mathrm{σ}$
 • The keyword argument indexlist = ind allows the user to specify the index structure for the bivector solder form. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form  is returned.
 • The bivector soldering forms satisfy a large number of identities, some of which are illustrated in Examples 2 - 4.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BivectorSolderForm(...) only after executing the commands with(DifferentialGeometry; with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-BivectorSolderForm.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

First create a vector bundle over with base coordinates  and fiber coordinates .

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Define a metric g on M. Note that our spinor conventions have the metric with signature

 > $g≔\mathrm{evalDG}\left(\mathrm{dt}&t\mathrm{dt}-\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}-\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right)$ (2.2)

Define an orthonormal frame on M with respect to the metric g.

 > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.3)

Calculate the solder form sigma from the frame F.

 > $\mathrm{σ}≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{5}{,}{8}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{6}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right)$ (2.4)

Calculate the bivector solder form S from sigma.

 > $S≔\mathrm{BivectorSolderForm}\left(\mathrm{σ},"spinor"\right)$
 ${S}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right)$ (2.5)

Example 2.

The contraction of two bivector solder forms on their tensor indices can be expressed in terms of the Kronecker delta spinor.

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

 > $\mathrm{S1}≔\mathrm{BivectorSolderForm}\left(\mathrm{σ},"spinor"\right)$
 ${\mathrm{S1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]\right]\right]\right)$ (2.6)
 > $\mathrm{S2}≔\mathrm{BivectorSolderForm}\left(\mathrm{σ},"spinor",\mathrm{indextype}=\left["con","con","cov","cov"\right]\right)$
 ${\mathrm{S2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right)$ (2.7)
 > $\mathrm{LHS}≔\mathrm{ContractIndices}\left(\mathrm{S1},\mathrm{S2},\left[\left[1,1\right],\left[2,2\right]\right]\right)$
 ${\mathrm{LHS}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right)$ (2.8)

To calculate the right-hand side we construct the symmetrized tensor product of 2 Kronecker delta spinors and multiply by 8 (because SymmetrizeIndices will include a factor of 1/2).

 > $\mathrm{δ}≔\mathrm{KroneckerDeltaSpinor}\left("spinor"\right)$
 ${\mathrm{δ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)$ (2.9)
 > $E≔\mathrm{RearrangeIndices}\left(\mathrm{δ}&t\mathrm{δ},\left[\left[2,3\right]\right]\right)$
 ${E}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)$ (2.10)
 > $\mathrm{RHS}≔8&mult\left(\mathrm{SymmetrizeIndices}\left(E,\left[1,2\right],"Symmetric"\right)\right)$
 ${\mathrm{RHS}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{8}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{4}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{4}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{8}\right]\right]\right]\right)$ (2.11)

Check that the LHS and RHS are the same.

 > $\mathrm{LHS}&minus\mathrm{RHS}$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_vrt"}{,}{"con_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (2.12)

Example 3.

The contraction of two bivector soldering forms on their tensor indices can be expressed in terms of the metric and the permutation tensor

.

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

 > $\mathrm{S3}≔\mathrm{BivectorSolderForm}\left(\mathrm{σ},"spinor",\mathrm{indextype}=\left["cov","cov","cov","cov"\right]\right)$
 ${\mathrm{S3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{1}{,}{4}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}{,}{6}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}{,}{5}{,}{6}\right]{,}{I}\right]{,}\left[\left[{2}{,}{3}{,}{6}{,}{5}\right]{,}{I}\right]{,}\left[\left[{2}{,}{4}{,}{5}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{4}{,}{6}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{1}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{3}{,}{1}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{3}{,}{2}{,}{5}{,}{6}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{2}{,}{6}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{5}{,}{5}\right]{,}{-I}\right]{,}\left[\left[{3}{,}{4}{,}{6}{,}{6}\right]{,}{I}\right]{,}\left[\left[{4}{,}{1}{,}{5}{,}{6}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}{,}{6}{,}{5}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{2}{,}{5}{,}{5}\right]{,}{1}\right]{,}\left[\left[{4}{,}{2}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}{,}{5}\right]{,}{I}\right]{,}\left[\left[{4}{,}{3}{,}{6}{,}{6}\right]{,}{-I}\right]\right]\right]\right)$ (2.13)
 > $\mathrm{LHS}≔\mathrm{ContractIndices}\left(\mathrm{S1},\mathrm{S3},\left[\left[3,3\right],\left[4,4\right]\right]\right)$
 ${\mathrm{LHS}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{-2}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{2}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{1}{,}{2}{,}{4}{,}{3}\right]{,}{2}{}{I}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{-2}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{4}\right]{,}{2}{}{I}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{2}\right]{,}\left[\left[{1}{,}{3}{,}{4}{,}{2}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{1}{,}{4}{,}{1}{,}{4}\right]{,}{-2}\right]{,}\left[\left[{1}{,}{4}{,}{2}{,}{3}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{1}{,}{4}{,}{3}{,}{2}\right]{,}{2}{}{I}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}{2}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{2}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{-2}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{4}\right]{,}{2}{}{I}\right]{,}\left[\left[{2}{,}{1}{,}{4}{,}{3}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{4}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{2}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-2}\right]{,}\left[\left[{2}{,}{3}{,}{4}{,}{1}\right]{,}{2}{}{I}\right]{,}\left[\left[{2}{,}{4}{,}{1}{,}{3}\right]{,}{2}{}{I}\right]{,}\left[\left[{2}{,}{4}{,}{2}{,}{4}\right]{,}{2}\right]{,}\left[\left[{2}{,}{4}{,}{3}{,}{1}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-2}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}{2}\right]{,}\left[\left[{3}{,}{1}{,}{2}{,}{4}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-2}\right]{,}\left[\left[{3}{,}{1}{,}{4}{,}{2}\right]{,}{2}{}{I}\right]{,}\left[\left[{3}{,}{2}{,}{1}{,}{4}\right]{,}{2}{}{I}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}{-2}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{2}\right]{,}\left[\left[{3}{,}{2}{,}{4}{,}{1}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{3}{,}{4}{,}{1}{,}{2}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{3}{,}{4}{,}{2}{,}{1}\right]{,}{2}{}{I}\right]{,}\left[\left[{3}{,}{4}{,}{3}{,}{4}\right]{,}{2}\right]{,}\left[\left[{3}{,}{4}{,}{4}{,}{3}\right]{,}{-2}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}{2}\right]{,}\left[\left[{4}{,}{1}{,}{2}{,}{3}\right]{,}{2}{}{I}\right]{,}\left[\left[{4}{,}{1}{,}{3}{,}{2}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}{-2}\right]{,}\left[\left[{4}{,}{2}{,}{1}{,}{3}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{4}{,}{2}{,}{2}{,}{4}\right]{,}{-2}\right]{,}\left[\left[{4}{,}{2}{,}{3}{,}{1}\right]{,}{2}{}{I}\right]{,}\left[\left[{4}{,}{2}{,}{4}{,}{2}\right]{,}{2}\right]{,}\left[\left[{4}{,}{3}{,}{1}{,}{2}\right]{,}{2}{}{I}\right]{,}\left[\left[{4}{,}{3}{,}{2}{,}{1}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{4}{,}{3}{,}{3}{,}{4}\right]{,}{-2}\right]{,}\left[\left[{4}{,}{3}{,}{4}{,}{3}\right]{,}{2}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{-2}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}{2}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{1}{,}{2}{,}{4}{,}{3}\right]{,}{2}{}{I}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{-2}\right]{,}\left[\left[{1}{,}{3}{,}{2}{,}{4}\right]{,}{2}{}{I}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}{2}\right]{,}\left[\left[{1}{,}{3}{,}{4}{,}{2}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{1}{,}{4}{,}{1}{,}{4}\right]{,}{-2}\right]{,}\left[\left[{1}{,}{4}{,}{2}{,}{3}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{1}{,}{4}{,}{3}{,}{2}\right]{,}{2}{}{I}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}{2}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}{2}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{-2}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{4}\right]{,}{2}{}{I}\right]{,}\left[\left[{2}{,}{1}{,}{4}{,}{3}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{2}{,}{3}{,}{1}{,}{4}\right]{,}{-}{2}{}{I}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{2}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}{-2}\right]{,}\left[\left[{2}{,}{3}{,}{4}{,}{1}\right]{,}{2}{}{I}\right]{,}\left[\left[{2}{,}{4}\right]\right]\right]\right]\right)$